Quantile hedging pension payoffs:

an analysis of investment incentives∗

Anne MacKay†

October 18, 2017

(Accepted version)

Abstract

Pension payoffs that depend on the term structure of interest rates can be superhedged

by trading only in the bond market, albeit at a possibly high initial price. Quantile hedging

techniques can be used to reduce the cost of the strategy if the investor is willing to accept

the risk that, with low probability, the hedging portfolio will not be sufficient to cover the

payoff. In this paper, we use results on partial hedging of defaultable claims to assess whether

the cost of a quantile hedging strategy can be reduced by the addition of a risky asset to the

bond market. In a simplified market model, we derive explicit expressions for the initial cost

of the hedge and the expectation of the unhedged loss. We show that while investing in the

risky asset reduces the cost of the hedge, it can significantly increase the risk linked to the

unhedged loss. In the context of pension plan funding, we thus provide an example of the

risky investment strategies encouraged by a partial hedge criterion based on the probability

of loss.

1 Introduction

We consider a bond market, in which we seek to hedge a payoff linked to the term structure

of interest rates and contingent on the survival of an individual. Setting up a superhedging

portfolio would ensure that there is always sufficient funds to pay the claim at maturity. The

initial cost of such a hedging strategy is often very high.

An investor may not always be willing to spend the full superhedging price on a hedging strategy.

In exchange for investing a smaller amount, she may accept to take some risks. Quantile hedging,

introduced by Follmer and Leukert (1999), describes hedging strategies that cover a payoff with

maximal probability, at a fixed cost which is lower than the superhedging price. In this paper,

we consider a pension payoff for which a quantile hedge can be constructed using only bonds,

for a cost lower than the cost of a superhedge. However, the question we seek to answer here

is whether adding a risky asset to the hedging portfolio can further reduce the initial cost of

∗The author acknowledges support from NSERC and from Risklab at ETH Zurich, where she was a postdoctoral

researcher when the first version of this article was written.†Deparment of Mathematics, Universite du Quebec a Montreal, Montreal, Canada.

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the quantile hedging strategy. We also want to highlight the possible risks involved with such a

hedging criterion.

When considering only superhedging strategies for the pension payoff, it will be clear that the

additional risky asset is useless to hedge a claim that only depends on the term structure of

interest rates. However, if the investor is willing to consider a hedge that only covers the payoff

with high probability (but still less than 1), is it possible to further reduce the price of the

replicating portfolio by also investing in equity?

This question is relevant to pension plan funding. Apart from demographic risks, the payoffs

linked to defined benefit plans or hybrid plan designs, such as cash balance (Hardy et al. Hardy,

Saunders, and Zhu (2014)), depend mostly on the evolution of the term structure of interest

rates. In particular, some cash balance plans credit guaranteed returns expressed as functions

of a spot rate.

In the industry, these payoffs are however often hedged using a portfolio which is significantly

invested in equity. This can be explained by a few different factors. First, while investments

in equity are generally riskier, they yield higher returns on average. This additional return,

compensating for the additional risk taken on by investing in equity, is called equity risk premium.

There is some evidence of its existence (Dimson, Marsh, and Staunton (2003)), but there is

no agreement on its size or its behaviour through time. Nonetheless, on longer horizons, it

is generally accepted that equity will yield a higher average return than bonds, which may

motivate the composition of pension funds. Second, since stock and bond prices tend to be

negatively correlated, particularly in bear markets (see, for example, Yang, Zhou, and Wang

(2009)), investing in equity may provide a hedge for interest rate linked payoffs. Therefore, it

is relevant to ask whether the pension industry’s investments in equity are supported by the

theory of quantile hedging.

It is also crucial to assess the risk resulting from a quantile-type hedging criterion. Indeed,

a criterion that incentivizes investment strategies that have a small probability of resulting

in a big loss might not be appropriate for the pension and life insurance industry, since it is

responsible for the financial security of a large number of retirees. Nonetheless, by focusing on

the probability of a loss, regulation based on risk measures such as the Value-at-Risk (VaR)

may create the same type of incentive. Therefore, the study of the investment incentives linked

to quantile hedging in the pension industry can also be a further example of potential problems

with regulation that only take the probability of loss into account.

General results on quantile hedging were first presented in Follmer and Leukert (1999). Sekine

(2000) and Nakano (2011) obtained results pertaining to the partial hedging of defaultable

claims. As those results also apply in the context of survival contingent payoffs, we review

them in this paper. Quantile hedging has been used to hedge different derivatives in various

markets. For example, Krutchenko and Melnikov (2001) develop explicit formulas to hedge

options in a jump-diffusion market. Many authors also use quantile hedging in the context of

life insurance. In particular, Melnikov and Skornyakova (2005) apply quantile hedging to pure

endowment equity-linked contracts, Gao, He, and Zhang (2011) generalize these results to a

bigger class of equity-linked insurance contracts, and Wang (2009) focuses on death benefits.

2

Melnikov and Tong (2012, 2013, 2014) study similar problems in markets with stochastic interest

rates and transaction costs. Also in life insurance, Klusik and Palmowski (2011) explore the

incompleteness brought on by insurance risks such as mortality.

Quantile hedging was extended to more general partial hedging strategies in Follmer and Leukert

(2000). In particular, these strategies take into account the size of the loss. In that mindset,

Barski (2016) provides a refinement of the strategy which also limits the loss, while working

with a slightly different definition of admissible strategies. Finally, Cong, Tan, and Weng (2013,

2014) also present hedging approaches based on the distribution of the unhedged loss. In this

paper, we focus mainly on the results of Follmer and Leukert (1999) and Sekine (2000), and

reserve the application of further partial hedging approaches for future work.

Most of the literature applying quantile hedging to life insurance is concerned with the risk

management of equity-linked payoffs by trading in a (more or less) general equity market. Our

approach differs in this way: the payoff we seek to hedge is only linked to interest rates, and

could be superhedged by only investing in bonds. However, we extend the market to include a

risky asset, and assess its impact on the cost of the strategy. Mathematically, we seek to hedge

the claim using a filtration that is richer than the sub-filtration on which the claim is defined. To

the author’s knowledge, this application of quantile hedging has not previously been explored.

An important motivation for our work is Hardy, Saunders, and Zhu (2014), which demonstrates

that the (risk-neutral) market value of a cash balance plan benefit is typically much higher than

the value actuaries assign to the payoff for plan funding purposes. In this paper, we attempt

to explain this seemingly underfunded position by two factors. First, we assume that the plan

sponsor is willing to take a small chance of a loss when the claim comes to maturity. Second,

we permit investment in a risky asset, which allows the hedger to make use of the equity risk

premium to reduce the cost of the hedge. These assumptions seem in line with the funding

practices of pension plan sponsors.

Our results show that, in the simplified market we use to derive explicit expressions, the avail-

ability of a risky asset is in fact effective in reducing the cost of a quantile hedge. However, we

also demonstrate that the optimal quantile hedge invested in part in equity can be significantly

riskier than its bond-only counterpart. Therefore, a hedging criterion defined in terms of the

probability of loss may give an incentive to implement riskier hedging strategies by reducing their

prices, which is not desirable in the pension industry, where the failure of a hedging strategy

can have a significant impact on the financial security of large groups of retirees.

To study the problem described above, we define two markets: in the first one, bonds are the

only assets available for trading. The second market is constructed by adding a risky asset to

the first one. Using quantile hedging results for defaultable claims presented in Sekine (2000),

we derive the cost of the optimal hedging strategy in each market.

In Section 2, we introduce the two markets in which we compare the cost of the quantile hedge,

and we present the payoff we seek to hedge. In Section 3, quantile hedging results are reviewed

and applied to our setting. This allows us to compare the costs and the riskiness of the quantile

hedge in both markets. Section 4 contains numerical illustrations and Section 5 concludes.

3

2 Setting

Our main goal in this paper is to compare the cost of a partial hedging strategy for an interest

rate linked payoff in two different market settings. In the first market, bonds are the only assets

available for trading, while in the second one, investors can also trade in a risky asset. In this

section, we present these two markets in further details, discuss mortality risk and introduce the

payoff we consider throughout the rest of the paper.

2.1 Financial markets

2.1.1 Bond-only market

We work on a probability space (Ω,G,P), where P denotes the real-world (objective) measure.

We also fix a time horizon T ∈ R+, which will represent the maturity of a payoff, and a final

time horizon 2T . We extend the time horizon beyond the maturity of the payoff to ensure that

the prices of bonds with time to maturity of at most T are well defined up to time T .

We consider a bond market in which at each time t, 0 ≤ t ≤ T , bonds of all times to maturity

τ ∈ [0, 2T − t] are traded. Then, from times 0 to T , it is always possible to trade bonds with

time to maturity T (thus coming to maturity at the latest at time 2T ). We denote the price at

time t of a bond maturing at s by P (t, s), for 0 ≤ t ≤ T , 0 ≤ s ≤ 2T with t ≤ s. To exclude

arbitrage, we assume that these prices are strictly greater than 0 with P (t, t) = 1, for 0 ≤ t ≤ T .

The short rate process r(t)0≤t≤T can be extracted from the bond market. To do so, we define

the instantaneous forward rate at time t for maturity s, denoted by f(t, s), by

f(t, s) := −∂ logP (t, s)

∂s,

under the assumption that bond prices are sufficiently smooth. The instantaneous forward rate

is the rate prevailing at time t for the infinitesimal time period [s, s+ ds), t ≤ s. The short rate

r(t) is then defined by

r(t) := f(t, t).

We also assume the existence of a bank account accumulating at the short rate. The value

process of this asset is denoted by B(t)0≤t≤T , so that

B(t) = e∫ t0 r(s)ds.

We let the short rate be modeled by an Ornstein-Uhlenbeck process:

dr(t) = a(b− r(t))dt+ σr dWr(t), (2.1)

for t ≥ 0, where a, b and σr are positive constants, and Wr = Wr(t)t≥0 denotes a P -Brownian

motion. This model is known in the literature as the Vasicek model (Vasicek, 1977). We further

define FB = FBt t≥0 as the filtration generated by Wr and augmented by the P -null sets of G.

We also assume that the market price of interest risk, denoted by θr, is unique and constant, so

that the price P (t, s) at time t of a bond with maturity s is given by

P (t, s) = expγ(s− t)− r(t)D(s− t), (2.2)

4

where

D(s− t) =1− e−a(s−t)

a,

γ(s− t) =

(b+

σrθra− σ2

r

2a2

)(D(s− t)− (s− t))− σ2

r

4a(D(s− t))2. (2.3)

This result can be found in numerous references, one of which is Brigo and Mercurio (2007).

It follows that under the P -measure, the bond price has the following dynamics:

dP (t, s)

P (t, s)= r(t)dt− σ(s−t)(dWr(t)− θr dt), (2.4)

where σ(s−t) = σrD(s− t). In particular, the negative sign in front of σ(s−t) highlights the fact

that bond prices rise when interest rates fall, and vice and versa.

For 0 ≤ t ≤ T , we define the process ZB = ZBt 0≤t≤T by

ZBt = eθrWr(t)− 12θ2r t, (2.5)

and observe that it is a strictly positive local martingale with E[ZBt]

= 1 for any 0 ≤ t ≤ T .

Therefore, we can define a new probability measure equivalent to P on (Ω,FB) by dPB

dP

∣∣FT

= ZBT .

It can be shown (for example, using Girsanov’s Theorem and (2.4)) that under PB, discounted

bond prices are local martingales. It follows that PB is an equivalent local martingale measure

(EMM).

By Theorem 4.9 of Filipovic (2009), since the market price of risk θr is unique, the EMM defined

by (2.5) is unique and our bond market is complete with respect to the filtration FB. It follows

in particular that any FBT -measurable claim with finite expectation can be perfectly replicated

by trading in the bond market. The initial cost of this strategy is given by the PB-expectation

of the discounted payoff, its unique no-arbitrage price.

Remark 1. Since we are working with a one-factor short-rate model, the assumption that bonds

of all maturities are available for trading at all times is not necessary. In fact, the bank account

and a bond with maturity 2T is sufficient to replicate all the other bonds in our market. There-

fore, going forward we can assume that those are the only two assets available for trading in the

bond market, without losing its completeness.

2.1.2 Mixed market

The main goal of this paper is to assess whether the availability of equity can reduce the cost

of imperfect hedging strategies for a pension-type claim, which is linked to the bond market

and paid if the plan member reaches a certain age. The financial part of the payoff will be

represented by an FBT -measurable random variable with finite expectation, and it will therefore

be possible to superhedge it by trading only in the bond market. Nonetheless, to account for

the possibility of investing in equity, we introduce a second market, called the mixed market,

which we obtain by adding a risky asset to the bond market, and by enlarging the filtration

appropriately. In Section 3, we will compare the cost of hedging strategies developed in each of

the two markets (bond-only and mixed).

5

The mixed market is also defined on the space (Ω,G, P ), and it is obtained by enlarging the

filtration FB as follows. Let S = St0≤t≤T denote the price process of the risky asset discounted

by the bank account numeraire, with P -dynamics given by

dS(t)

S(t)= σ1 (dW (t) + θ dt)− σ2 (dWr(t)− θr dt) , (2.6)

where θ, σ1 and σ2 are constants with σ1 6= 0. θr is the market price of interest rate risk defined

in Section 2.1.1. The process W = W (t)t≥0 denotes a Brownian motion independent of Wr.

If σ2 6= 0, then the risky asset discounted by the bank account numeraire is correlated to the

interest rate through the process Wr. Note that σ2 can be either positive or negative. When σ2

is positive, then S is positively correlated with bond prices (this can easily be seen by comparing

(2.4) and (2.6)).

We let FS = FSt 0≤t≤T denote the filtration induced by W and augmented by the P -null

sets of G. Finally, we can define FM as the filtration generated by the mixed market, with

FMt = FBt ∨ FSt , 0 ≤ t ≤ T .

For 0 ≤ t ≤ T , we define the process ZM = ZMt 0≤t≤T by

ZMt = eθrWr(t)−θW (t)− 12

(θ2r+θ2)t. (2.7)

Since the process defined in (2.7) is a strictly positive local martingale with E[ZMt

]= 1 for

0 ≤ t ≤ T , it can be used to define a new probability measure PM equivalent to P on (Ω,FM )

by dPM

dP

∣∣FT

= ZMT . Simple calculations show that under this new measure, the discounted bond

and risky asset price processes are local martingales. Therefore, PM is an EMM.

For the filtration FM , which is induced by the two-dimensional Brownian motion (Wr,W ), the

market price of risk vector (θr, θ) is unique. It follows (by Theorem 4.9 of Filipovic (2009))

that PM is the unique EMM for the mixed market, and that the market is complete. As an

immediate consequence, any FMT -measurable payoff is perfectly replicable by trading in the bank

account, the bond and the risky asset. The initial cost of the replicating portfolio is given by

the unique no-arbitrage price obtained by taking the PM -expectation of the discounted payoff.

2.1.3 General notation for the financial market

In Section 3, we will be interested in comparing partial hedging strategies developed in both the

bond-only and the mixed market. As will be shown in the next section, the quantities of interest

will be functions of the density of the unique EMM, namely ZBT in the bond-only market and

ZMT in the mixed one. In order to simplify the exposition of the results, we will work with a

general ZT of the form

ZT := exp

θ′W (T )− T

2θ′θ

, (2.8)

where θ = (θ1, θ2)′, with θ1, θ2 ∈ R, and W (T ) = (Wr(T ),W (T ))′, and we will denote by P ∗

the EMM defined by ZT = dP ∗

dP

∣∣FT

.

To obtain specific results for the bond-only market, it will suffice to let θ = (θr, 0)′, in which

case ZT will be equal to (2.5). For the mixed market, letting θ = (θr,−θ)′ will yield the desired

6

result. Going forward, we denote

θ(B) = (θr, 0)′

θ(M) = (θr,−θ)′.

Therefore, in the rest of this section and in most of Section 3, we will work with general ZT ,

θ and W , only to replace them with market-specific values to compare the partial hedging

strategies in each market. We also use F = Ft0≤t≤T to denote either FB ou FM , the filtration

induced by the financial market.

2.2 Mortality risk

The payoff we seek to hedge in this paper is a simplified version of the payout of a pension

plan. As such, it is only paid if the employee survives until retirement. We therefore need

to incorporate mortality risk in our framework. To model the time of death of the employee

receiving the payout, we borrow the reduced-form framework from credit risk theory (see, for

example, Bielecki and Rutkowski (2013)). This will allow us to use the results of Sekine (2000)

and of Nakano (2011) on quantile hedging for defaultable securities.

We let τ be a positive random variable denoting the random time of death, with P (τ > t) > 0

for any 0 ≤ t ≤ T , and define Nt0≤t≤T , with Nt = 1τ≤t. We denote by H = Ht0≤t≤T the

filtration induced by Nt, with

Ht := σNs; 0 ≤ s ≤ t,

and let G := Gt0≤t≤T , with Gt = Ft ∨Ht.We further define the survival process Λt0≤t≤T with respect to Ft as

Λt := P (τ > t|Ft),

and assume that Λt > 0 for 0 ≤ t ≤ T . We let the process λt0≤t≤T be defined implicitly by

Λt = e−∫ t0 λs ds.

It follows that λt is Ft-adapted. Throughout the paper, we use the following assumption.

Assumption 1. The survival process Λt is independent of the financial market filtration F.

It follows from Assumption 1 that Λt = P (τ > t), for 0 ≤ t ≤ T .

We further define

Mt = Nt −∫ t

0λs(1−Ns−)ds, 0 ≤ t ≤ T,

which can be shown to be a G-martingale (see Bielecki and Rutkowski (2013)). We also let

Zδ = Zδt 0≤t≤2T be defined by

Zδt := (1 + δt1τ≤t) exp

(∫ t∧T

0δsλsds

),

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where δ = δt0≤t≤2T is an element of

D = δ : bounded, G-predictable, δ > −1 dt× dP a.e.

It is possible to show that for δ ∈ D, Zδ satisfies

Zδt = 1 +

∫ t

0δsZ

δsdMs,

for 0 ≤ t ≤ T , and is thus a (G, P )-martingale (see for example Bielecki and Rutkowski (2013)).

Finally, we have that both Zt0≤t≤T , with

Zt = exp

θ′W (t)− t

2θ′θ

,

and ZtZδt 0≤t≤T are (G, P )-positive martingales for any δ ∈ D. Furthermore, since Zδ is

orthogonal to any (F, P )-martingales for any δ ∈ D, the discounted price processes of the assets

traded in the financial market are martingales under the measure P δ defined by dP δ

dP

∣∣GT

= ZTZδT .

It follows that

P := P δ : δ ∈ D

defines the class of equivalent martingale measures on G. Note that in particular, P contains

P ∗, the unique EMM for the purely financial market defined by the filtration F. This can be

shown by letting δ ≡ 0 in (2.2). For more details on this setting, see Bielecki and Rutkowski

(2013).

While the purely financial markets described in Section 2.1 were both complete, the addition of

mortality renders incomplete the general market combining financial and mortality risk. That

is, GT -measurable payoffs are not necessarily perfectly replicable by trading only in the available

assets. For more details on incomplete markets, see for example Karatzas and Shreve (1998).

2.3 Hedging in incomplete markets

A portfolio strategy is characterized by its initial capital V0 and by a Gt-predictable process

ξ = ξt0≤t≤T that describes the way the portfolio is invested in the different traded assets. A

portfolio strategy V0, ξ is called admissible if its (discounted) value process1

Vt = V0 +

∫ t

0ξsdXs

satisfies

Vt ≥ 0, for all 0 ≤ t ≤ T, P -a.s.

Here we use X = Xt0≤t≤2T to denote the possibly multi-dimensional discounted price process

of the assets traded on the financial market. That is, in the bond-only market, X denotes

1Throughout the paper, similarly to Follmer and Leukert (1999), we consider payoffs and asset prices discounted

by the bank account numeraire. This simplifies the notation and does not affect the comparison between the two

markets.

8

the (discounted) price process of the bond with maturity 2T . In the mixed market, X is two-

dimensional and denotes the (discounted) price process of the bond and the risky asset.

It is possible to show (see Nakano (2011)) that the process ZtZδt Vt0≤t≤T is a supermartingale

when Vt0≤t≤T is the (discounted) value process resulting from an admissible strategy.

We consider a (discounted) GT -measurable payoff H, with

supδ∈D

E[ZTZδTH] <∞.

Since H is GT -measurable, it can depend on both the financial market and the survival process,

and it is therefore not necessarily possible to replicate it perfectly by trading in the market. A

so-called superhedging strategy ensures that VT ≥ H, P -a.s. It is the most conservative way to

hedge the payoff H. The superhedging cost for the payoff H, denoted by Π(H), is defined as

Π(H) = infx ≥ 0 : x+

∫ T

0ξs dXs ≥ H, P -a.s. for some admissible strategy ξ.

Simple expressions for the superhedging cost of a general claim H ∈ GT do not always exist.

However, it is possible to show that for a GT -measurable payoff H of the form

H = Y 1τ>T, (2.9)

with Y ∈ FT , the superhedging strategy for H coincides with the perfect replicating portfolio for

Y (see Proposition 2.1 of Nakano (2011)). Since we will use a version of this result throughout

the paper, we present it next. Here, E∗[·] denotes the expectation taken under P ∗, the unique

EMM in the financial market.

Proposition 2.1 (Proposition 2.1 of Nakano (2011)). Let H be defined as in (2.9) and assume

E∗[Y ] <∞. Then we have

Π(H) = E∗[Y ],

and the replicating portfolio for Y is the superhedging portfolio for H.

The proof of Proposition 2.1 is given in Nakano (2011). For completeness, we present it in

Appendix A.

2.4 Cash balance payoff

In this paper, we consider the payoff of a cash balance pension plan, which depends on the term

structure of interest rates and on the survival of an individual. Specifying the form of the claim

allows us to obtain analytic expressions for the cost of a quantile hedge. By doing so for each

market introduced in Sections 2.1.1 and 2.1.2, we can analyse the impact of the availability of

a risky asset on the initial cost of the partial hedge, and thus infer on the incentives created by

the quantile hedging criterion in the context of pension plans.

The claim we consider is inspired by a type of hybrid pension plan, and is general enough to

cover different types of pension benefits. Nonetheless, the form we use is still restrictive, and

9

the analysis we perform here could also be modified and applied to other types of interest rate

derivatives.

A cash balance plan is similar to a defined contribution plan, but the interest rate credited on

the contributions deposited in the employee’s account is defined by a pre-determined formula.

Cash balance plans are presented in greater details in Hardy, Saunders, and Zhu (2014). The

payoff they consider consists of the accumulation up to time T of an initial investment ζ0 at

a guaranteed, pre-determined rate. In this paper, we also take into account the fact that the

benefit is paid only if the employee is still alive at T . We however ignore any mortality benefits

that could be paid if the plan member does not survive until retirement

Of particular interest is the case where the crediting rate is linked to the term structure of

interest rates. Going forward, we follow Hardy, Saunders, and Zhu (2014) and assume that the

crediting rate on an initial contribution ζ0, denoted by rc(t;n), is given by

rc(t;n) := yn(t) + g,

for n > 0 and 0 ≤ t ≤ T , where g ≥ 0 is a constant and yn(t) is the yield to maturity at time t

of a bond with time to maturity n. In our setting, n is usually an integer representing a number

of years. For n > 0, this yield is defined by

yn(t) := − 1

nlog(P (t, t+ n)) =

1

n(r(t)D(n)− γ(n)).

The second equality stems from (2.2), and γ(n) and D(n) are defined as in (2.3). Note that

since γ(n) and D(n) only depend on n, the yield rate yn(t) only depends on t through r(t).

We also wish to consider constant crediting rates. To denote this case, we write rc(t; 0) = g.

Therefore, the case rc(t;n) = g will be denoted by n = 0.

Examples of such crediting rates include a pension plan that credits the current 3-year yield

rate, plus 0.5%, or the current 10-year yield rate. Letting n = 0 leads to the case where a

constant rate is guaranteed.

The payoff we consider is thus the accumulation at rate rc of the initial investment ζ0. We

denote the discounted financial payoff by ζ. It represents the amount paid given that the

employee survives to T , and it is given by:

ζ = ζ0e∫ T0 (rc(t;n)−r(t))dt

=

ζ0egT+

∫ T0 (yn(t)−r(t))dt, n > 0,

ζ0egT−

∫ T0 r(t)dt, n = 0.

(2.10)

Without loss of generality, we let ζ0 = 1 from now on.

Since the plan pays out only if the plan member survives to T , the (discounted) payoff we seek

to hedge, denoted by HT , is defined as

HT := ζ1τ>T = ζ(1−NT ). (2.11)

Remark 2. Our definition of HT simplifies the total payoff of a cash balance plan in different

ways. First, it does not take into account ongoing contributions made to the account. These

10

future contributions can be treated separately in a similar manner, or they can be included in

the initial amount at the next valuation date. Second, as in Hardy, Saunders, and Zhu (2014),

we ignore any benefits that could be paid should the plan member die before T .

Remark 3. Note also that by letting n = g = 0 and by using τ to denote a default time, HT is

the discounted payoff of a defaultable zero-coupon bond with maturity T . Quantile hedging a

defaultable zero-coupon bond is thus a special case of the results that follow. Our framework

can also be used to model the payoff of a defined benefit plan, whose payout at T is known

(under certain assumptions on salary growth and mortality) at time 0. Therefore, our analysis

can be extended beyond cash balance plans.

Note that the pure financial payoff ζ is both FT - and GT -measurable, while the payoff HT is

only GT -measurable. Since the pure financial market is complete, ζ can be perfectly replicated

by trading in the available assets. This is however not the case for HT . We have the following

results on the superhedging cost of HT .

Corollary 2.2. Let HT be as in (2.11). Then, we have

Π(HT ) = E∗[ζ], (2.12)

where E∗[·] denotes the expectation taken under the P ∗-measure. Furthermore, the replicating

portfolio for ζ is the superhedging portfolio for HT .

Proof. Corollary 2.2 follows immediately from Proposition 2.1.

It our market model, it is possible to obtain the density of ZT ζ under P and to derive an analytic

expression for E∗[ζ]. We present those results here, since they will be used throughout the rest

of the paper in the context of quantile hedging.

Proposition 2.3. Let ζ be the payoff defined by (2.10) and let

ZT = exp

θ′W (T )− T

2θ′θ

.

Then, under P , ZT ζ follows a lognormal distribution with parameters(η(n, T )− T

2 θ′θ, ϑ2(n, T ;θ)

), where

η(n, T ) =

gT − γ(n)Tn + D(n)

((r0 − b)1−e−aT

a + bT), n > 0,

gT −(

(r0 − b)1−e−aTa + bT

), n = 0,

ϑ2(n, T ;θ) =D2(n)σ2

r

a2

(T −D(T )− a

2D2(0, T )

)− 2θrσrD(n)

a(D(T )− T ) + θ′θT,

and

D(n) =

D(n)−n

n , n > 0

−1, n = 0.

11

The proof of Proposition 2.3 is given in Appendix B. Proposition 2.3 allows us to calculate

E∗[ζ], which is presented next.

Corollary 2.4. Let ζ and ZT be defined as in Proposition 2.3. E∗[ζ] is then given by

E∗[ζ] = eη(n,T )−T2θ′θ+ 1

2ϑ2(n,T ;θ).

Proof. Note that E∗[ζ] = E [ZT ζ]. The result follows directly from the distribution of ZT ζ given

in Proposition 2.3. 2

Remark 4. The unique no-arbitrage price of the payoff ζ is the same in both the bond-only

market and the mixed market. Heuristically, this is due to the fact that the perfect replicating

strategy that underlies the no-arbitrage price only involves investing in bonds, whether or not

stocks are available. Mathematically, using the independence of W (t) and Wr(t), we have

E[ZMT ζ

]= E

[e−θW (T )− 1

2θ2TZBT ζ

]= E

[e−θW (T )− 1

2θ2T]E[ZBT ζ

]= E

[ZBT ζ

].

3 Quantile hedging the cash balance payoff

Corollary 2.2 shows that it is possible to build a superhedging portfolio that can cover the claim

HT with probability 1. The initial cost of such a strategy is the unique no-arbitrage price of

the financial claim ζ. For different reasons, a pension plan sponsor may not be willing to invest

the full market value of ζ in a hedging strategy, and is therefore exposed to the risk that the

value at T of the portfolio is not sufficient to cover the pension payout. In this context, for

a given initial portfolio value, what is the probability that the final value of the strategy is at

least equal to the payout? What is the cost of a hedging strategy that covers the payout with a

given probability? Are these quantities different when investment in equity is allowed? In this

section, we derive analytical results that allow us to compare the cost and effectiveness of such

quantile hedging strategies in both markets presented in Section 2.1.

More precisely, we consider an investor (or plan sponsor) who wants to set up a self-financing

hedging portfolio to that will maximize the probability of VT , the final value of the trading

strategy, being at least equal to HT , given that the initial cost V0 is at most a given threshold

V0. This corresponds to finding the admissible strategy V0, ξ that maximizes

P [VT ≥ HT ],

under the constraint V0 ≤ V0, where V0 < Π(HT ). We refer to this optimization problem as

(OP1).

We are also interested in the case where a hedger wants to find the optimal hedging strategy

that will cover the claim HT with a probability of at least 1− ε, at a minimal cost. Therefore,

12

we are looking for the admissible strategy V0, ξ that minimizes V0 under the constraint

P

[V0 +

∫ T

0ξs dXs ≥ HT

]≥ 1− ε.

If the initial cost of the strategy is equal to Π(HT ), then superhedging is possible. Thus, we

expect that the value V0 that solves the minimization will be strictly less than Π(HT ) for ε > 0.

We denote this optimization problem (OP2).

Such hedging strategies were considered in a general semi-martingale settings in Follmer and

Leukert (1999). In this section, we apply the results of Sekine (2000) and Nakano (2011) on

quantile hedging of defaultable claims, in order to study the resulting investment incentives

related the pension payoff presented in Section 2.4. We first provide results using the “general

market” notation introduced in Section 2.1.3. Market-specific notation will then be used to

analyse the impact of the availability of a risky asset on the initial cost of the hedge. We also

assess the riskiness of the resulting hedge.

Throughout the section, we mostly focus on minimizing the initial cost (OP2), since we believe

it better represents the preoccupations of a pension plan sponsor. Nonetheless, we also present

results on the maximal probability of success for a given initial cost (OP1).

3.1 Minimizing the initial cost of the hedging strategy

We first consider (OP2), that is, we seek to minimize the cost of a partial hedging strategy,

given that the pension payout is covered by the hedging portfolio with a given probability.

Theorem 3.1 (Theorem 2(A) of Sekine (2000)). Assume that there exists k∗ = k∗(ε) satisfying

E[1A1(k∗)(1− ΛT ) + 1A2(k∗)] = 1− ε,

with A1(k) = ΛT ≤ kZT ζ and A2(k) = ΛT > kZT ζ. Then the superhedging strategy of the

claim

HT = 1A2(k∗)ζ(1−NT )

is a solution of (OP2).

Proof. Theorem 3.1 is a special case of Theorem 2(A) of Sekine (2000), which pertains to a

defaultable claim with partial recovery d if default occurs before T . We obtain Theorem 3.1 by

setting d = 0.

By Theorem 3.1, the initial cost of the quantile hedging strategy is given by the superhedging

price of the modified payoff 1A2(k∗)ζ(1 − NT ). We give an explicit expression for this price in

the following proposition. Throughout the paper, Φ(·) denotes the standard normal distribution

function and zα = Φ−1(α) is the α-quantile of the standard Normal distribution.

Proposition 3.2. Let HT and ZT be defined by (2.11) and (2.8), respectively, and let Vt0≤t≤Tdenote the value process of an admissible hedging strategy. The minimal cost V0 of the quantile

hedging strategy that satisfies

P [VT ≥ HT ] ≥ 1− ε

13

is given by

E∗[ζ1A2(k∗)

]= E∗ [ζ] Φ (z1−ε? − ϑ(n, T ;θ)) , (3.1)

where ϑ2(n, T ;θ) is as defined in Proposition 2.3 and ε? = εΛT

.

Proof. This result is an application of Theorem 3.1. The minimal cost of the strategy corresponds

to the superhedging price of the modified payoff HT = 1A2(k∗)ζ(1−NT ).

To solve for k∗, observe that

E[1A1(k)(1− ΛT ) + 1A2(k)] = (1− ΛT ) + P (ζZT < kΛT )ΛT .

From the distribution of ζZT , it is clear that k∗ exists. Using Proposition 2.3, we obtain

k∗ = ΛT exp

−(η(n, T )− T

2)θ′θ − z1−ε?ϑ(n, T ;θ)

, (3.2)

with ε? = εΛT

. It follows that

A2(k∗) =

ZT ζ < exp

η(n, T )− T

2θ′θ + z1−ε?ϑ(n, T ;θ)

,

and thus 1A2(k∗)ζ ∈ FT . By Proposition 2.1, the superhedging portfolio for HT is the replicating

portfolio for 1A2(k∗)ζ, so that the minimal cost V0 is given by

E∗[ζ1A2(k∗)] = E[ZT ζ1ZT ζ<exp(η(n,T )−T

2θ′θ+z1−ε?ϑ(n,T ;θ))

].

The result follows from the distribution of ZT ζ. 2

As expected, the initial cost of the quantile hedging strategy given in Proposition 3.2 is bounded

above by E∗[ζ], the superhedging price of the original payoff HT .

3.1.1 Comparison of the initial hedging cost in the bond-only and the mixed mar-

ket

Proposition 3.2 gives an explicit expression for the minimal cost of a quantile hedging strategy

with probability of success 1−ε using the general financial market notation introduced in Section

2.1.3. Therefore, this result applies to both the bond-only and the mixed market; to get market-

specific expressions, it suffices to replace the general θ by θ(B) or θ(M) appropriately. This allows

us to compare the cost of the optimal quantile hedging strategy in the two markets presented

in Section 2.1.

Using market-specific notation, we have

EB[ζ1A2(k∗;B)

]= EB [ζ] Φ(z1−ε? − ϑ(n, T ;θ(B))) (3.3)

and

EM[ζ1A2(k∗;M)

]= EM [ζ] Φ(z1−ε? − ϑ(n, T ;θ(M))), (3.4)

where EM [·] and EB[·] denote the expectations taken under the measures PB and PM , respec-

tively.

14

As explained in Remark 4, the unique no-arbitrage price of the payoff ζ is the same in both

financial markets, that is, EB [ζ] = EM [ζ] . Therefore, being able to invest in equity only affects

the initial cost of the hedging strategy through the P ∗-variance parameters of the original payoff,

ϑ2(n, T ;θ(B)) and ϑ2(n, T ;θ(M)). From the definition of ϑ2(n, T ;θ), we have

ϑ2(n, T ;θ(M)) = ϑ2(n, T ;θ(B)) + θ2T

> ϑ2(n, T ;θ(B)) (3.5)

when θ 6= 0 (that is, when the market price of equity risk differs from 0). In that case,

ϑ(n, T ;θ(M)) > ϑ(n, T ;θ(B)). The monotonicity of Φ(·) simplifies the comparison between the

markets. We present this result in Corollary 3.3.

Corollary 3.3. Let EM [·] and EB[·] denote the expectations taken under the measures PB and

PM , respectively, and let

A2(k∗;B) =ZT ζ < eη(n,T )−T

2(θ(B))′θ(B)+z1−ε?ϑ(n,T ;θ(B))

A2(k∗;M) =

ZT ζ < eη(n,T )−T

2(θ(M))′θ(M)+z1−ε?ϑ(n,T ;θ(M))

.

If the market price of equity risk θ 6= 0, then

EM[ζ1A2(k∗;M)

]< EB

[ζ1A2(k∗;M)

],

where ζ is the payoff defined in (2.10).

Corollary 3.3 highlights the fact that when the market price of equity risk is different from zero,

being able to invest in the equity market can reduce the initial cost of a self-financing quantile

hedging strategy.

The application of Proposition 3.2 to each market also gives further insight into the characteris-

tics of the mixed market that influence the cost of the hedging strategy. In particular, σ2, which

characterizes the dependence between the randomness of the stock price and the randomness

of the interest rate, does not appear directly in (3.1). However, the market price of equity

risk θ, which appears only in θ(M), is linked to the ratio of the drift of the risky asset to its

volatility, both of which are functions of σ2. Therefore, the dependence between the risky asset

and the interest rate will impact the cost of the hedge through the market price of equity risk.

The availability of the risky asset reduces the cost of the hedging strategy in comparison to a

bond-only hedge, as long as the market price of equity risk is different from 0. The dependence

between the risky asset and the bond should also impact the quantile hedging strategy.

Expressing the cost of the quantile hedge obtained in Proposition 3.2 using the parameters

specific to each market, as it is done in the proof of Corollary 3.3, also highlights the impact of

the maturity T on the hedging cost. From (3.3) and (3.4), it is clear that the only difference in

the hedging cost comes from the P ∗-variance of the original payoff. Then, writing ϑ2(n, T ;θ(M))

in terms of ϑ2(n, T ;θ(B)) as in (3.5) shows that as the maturity of the payoff T increases, the

difference between the initial hedging cost in the bond-only and the mixed market grows. This

15

indicates that the equity risk premium has a more significant impact over a longer period of

time.

Finally, one would expect the cost of the hedging portfolio to increase to the superhedging

price of the payoff HT , given in (2.12), as ε goes to 0, regardless of the available assets. Since

limx→∞Φ(x) = 1, (3.3) and (3.4) satisfy this requirement. In Section 4, these last observations

are illustrated with a numerical example.

3.1.2 Unhedged payoff

The quantile hedge whose cost was derived in the previous section is only a partial hedge. On

the set Ac2(k∗)∩τ > T, where Ac2(k∗) is the complement of A(k∗), the payoff is not replicated,

and the hedger incurs a loss. We derive here the expected value of this loss, given that it is

strictly positive.

For a given probability 1 − ε of a successful hedge, we showed in the previous section that the

availability of equity can reduce the price of the replicating portfolio for the modified claim

ζ1A2(k∗). We are now interested in the average loss incurred by the pension plan sponsor given

that the hedge fails.

As in Section 3.1.1, we first derive an expression using the general financial market notation;

that is, we use the Radon-Nikodym derivative given by (2.8). The result thus obtained will then

apply to both the bond-only and the mixed market. To obtain market-specific expressions, it

will suffice to replace θ by θ(B) or θ(M) appropriately.

Using the general financial market notation, we define the loss at maturity L by L := HT −VT , where VT is the value at T of the superhedging portfolio for the modified payoff HT =

1A2(k∗)ζ(1 −NT ) as in Theorem 3.1.2 Therefore, the loss is equal to the full financial payoff ζ

on Ac2(k∗) ∩ τ > T. The next proposition gives an expression for its P -expected value, given

that the loss is positive. It is the (discounted) amount that the investor can expect to lose on

average when a loss occurs.

Proposition 3.4. Let VT be the value at time T of the superhedging portfolio for HT and let

L = HT − VT be the loss at maturity resulting from a quantile hedge that has a probability of

success of at least 1− ε. The expected value of L conditional on L > 0 is given by

E[L|L > 0] = (ε?)−1eη(n,T )+ 12ν2s2IΦ

(ν2s2

I + νµI|W θrT

ϑ(n, T ;θ)− z1−ε?

), (3.6)

where ε? = εΛT

, ν = D(n)σre−aT and s2

I = s2I|W + µI|WT , with

µI|W = −eaT

aT(D(T )− T )

s2I|W =

e2aT

a2

(D(T )−D2(0, T )

(a

2+

1

T

)).

2Throughout this work, we consider payoffs discounted with the bank account numeraire, so that the quantity

we are studying here is in fact the discounted loss.

16

The proof of Proposition 3.4 is given in Appendix C.

Proposition 3.4 shows that E[L|L > 0] only depends on the availability of the risky asset through

the P ∗-standard deviation ϑ(n, T ;θ). In order to compare the loss resulting from the unhedged

part of the payoff in each market, we replace the general market price of risk vector θ in (3.6)

by a specific one, θ(B) or θ(M). We also let the subscripts (B) and (M) identify the market in

which the quantile hedging strategy is derived.

Expressing the result of Proposition 3 for each specific market gives

E[L(B)|L(B) > 0] = (ε?)−1eη(n,T )+ 12ν2s2IΦ

(ν2s2

I + νµI|W θrT

ϑ(n, T ;θ(B))− z1−ε?

)(3.7)

and

E[L(M)|L(M) > 0] = (ε?)−1eη(n,T )+ 12ν2s2IΦ

(ν2s2

I + νµI|W θrT

ϑ(n, T ;θ(M))− z1−ε?

). (3.8)

From (3.5), we know that ϑ(n, T ;θ(M)) > ϑ(n, T ;θ(B)) when θ 6= 0. Therefore, the impact of

the availability of the equity depends on the sign of the numerator ν2s2I + νµI|W θrT in (3.7) and

(3.8). This is formalized in the following corollary.

Corollary 3.5. Let E[L(B)|L > 0] and E[L(M)|L > 0] be the expected conditional loss in the

bond-only and the mixed market, respectively, as given (3.7) and (3.8). Then, we haveE[L(M)|L(M) > 0] < E[L(B)|L(B) > 0], if θr < κ,

E[L(M)|L(M) > 0] ≥ E[L(B)|L(B) > 0], if θr ≥ κ,

where κ = − νs2IµI|WT .

Proof. The proof follows from the definition of ν, s2I and µI|W in Proposition 3.4. Note that it

is possible to show that ν < 0 for all a, T > 0 and n ≥ 0. 2

The previous result is presented in terms of θr for ease of interpretation. The idea behind

Corollary 3.5 is that when the market price of interest rate risk is too low, having the opportunity

to invest in the risky asset can reduce the risk (expressed in terms of the unhedged loss) of the

strategy. In Section 5, we explore this result further through numerical examples.

The next proposition qualifies the behaviour of the expected loss as the probability of loss ε

goes to 0. We use the general notation L to refer to the loss associated with quantile hedging

in either market, L(B) or L(M), since the result is the same in both cases.

Proposition 3.6. Let L = HT − VT , and denote κ = − νs2IµI|WT . Then,

limε→0

E[L|L > 0] =

∞, θr < κ,

ΛT eη(n,T )+ 1

2ν2s2I , θr = κ,

0, θr > κ.

Proof. The proof follows from Lemma D.1, presented in Appendix D. 2

17

We showed in the previous section that as ε decreases to 0, the price of the hedging strategy

approaches the superhedging price of the payoff. Intuitively, one may imagine that as the hedging

strategy approaches perfect replication, E[L|L > 0] would go to 0, whether investing in equity is

permitted or not. However, this is not necessarily true, as shown in Proposition 3.6. In fact, the

expectation of the loss only goes to 0 when ν2s2I + νµI|W θrT is negative, that is, when θr > κ.

In Section 5, we present numerical examples in the case where this condition is not satisfied.

Proposition 3.4 highlights one of the risks linked to the use of the probability of loss as a hedging

criterion under such market conditions.

The positive probability of an unbounded loss resulting from quantile hedging strategies for cer-

tain parameter sets has already been discussed in the literature, in particular in the introduction

of Barski (2016). It is one of the pitfalls of classical quantile hedging and represents a reason

why the probability of a loss is not necessarily an appropriate hedging criterion, especially for

pension payoffs. This shortcoming of quantile hedging has motivated further work on partial

hedging, for example in Barski (2016); Follmer and Leukert (2000). The application of these

results to our setting is out of the scope of this paper.

One important remark is that the limiting behaviour of the expected loss does not depend

on the market (bonds-only or mixed) in which the claim is hedged. As previously explained,

Proposition 3.6 uses the general notation L because it holds for the loss in the bond-only market

L(B) as well as for the one in the mixed market L(M). One can thus conclude that if the

expected loss is unbounded when the partial hedging portfolio is partially invested in equity, it

has the same limiting behaviour in the bond-only market. While the size of the loss might differ,

unboundedness of the expected loss as ε approaches 0 is therefore only a result of the structure

of the bond part of the market.

3.2 Maximizing the probability of success

The first part of Section 3 focuses on the minimal hedging cost for a given probability of success,

because we believe that it better represents the problem faced by a pension plan sponsor. In

fact, it seems reasonable to set a maximal probability of financial loss, and to fund the plan

accordingly. Nonetheless, we also present here an explicit expression for the maximal probability

of success of a quantile hedge given an upper bound on its initial cost. This is the problem we

refer to as (OP1).

Theorem 3.7 (Theorem 1(A) of Sekine (2000)). Assume that there exists k? = k?(V0) satisfying

E∗[ζ1A(k?)] = V0,

with A(k) = ΛT > kZT ζ. Then the superhedging strategy of the modified claim HT = 1A(k?)ζ(1−NT )

is a solution of (OP1).

Proof. Theorem 3.7 is a special case of Theorem 1(A) of Sekine (2000), which studies defaultable

claims with partial recovery d. Theorem 3.7 is obtained by letting d = 0.

Below, we present an explicit expression for the maximal probability of success of the quantile

hedge subject to the initial cost being at most V0.

18

Proposition 3.8. Let HT and ZT be defined by (2.11) and (2.8), respectively, and let Vt0≤t≤Tdenote the value process of an admissible hedging strategy. The maximal probability P (VT ≥ HT )

subject to V0 ≤ V0 is equal to

(1− ΛT ) + ΛTΦ

(Φ−1

(V0

E∗[ζ]

)+ ϑ(n, T ;θ)

).

Proof. This result is an application of Theorem 3.7. The optimal set A(k?) is defined by

E∗[1A(k?)] = V0, which can be re-written as

E[ZT ζ1ZT ζ<ΛT k−1

]= V0.

Using the distribution of ZT ζ given in Lemma 2.3 yields k?. The optimal strategy is thus given

by the superhedging portfolio for 1A(k?)ζ(1−NT ), which coincides with the replicating strategy

for 1A(k?)ζ, by Proposition 2.1. Using the independence between the survival process and the

financial market, we can write the probability of success as

P (VT ≥ HT ) = P (1A(k?)ζ ≥ HT )

= (1− ΛT ) + ΛTP (A(k?)).

The result follows from the distribution of ZT ζ. 2

4 Numerical Illustrations

In this section we complement the analysis of the results of Propositions 3.2 and 3.6 with some

numerical illustrations.

For this purpose, we use the following parameters:

Table 1: Market Parameters.

Interest rate Equity

a = 0.035 σ1 = 0.18

b = 0.02 σ2 = 0.05

σr = 0.008 θ = 0.24

r0 = 0.02

θr = 0.12

Our market model is too simple to provide a good fit to market data. Nonetheless, we choose

a parameter set that is representative of typical market conditions. Using the interest rate

parameters of Table 1, the long-term unconditional mean for r(t) is 0.02, while its long term

unconditional standard deviation is 0.03024. These values are in line with those obtained by

Hardy, Saunders, and Zhu (2014). The equity parameters yield log-returns with a standard

deviation of 0.1868. The drift of the log-return on equity over the risk-free rate is 0.03175.

19

We consider a cash balance payoff with maturity T = 20 years, unless otherwise stated, where

the constant part of the crediting rate is equal to g = 0.01 and the variable part is linked to the

10-year spot rate (such that n = 10). We still assume that ζ0 = 1.

For the probability of survival to maturity ΛT = P (τ > T ), we use the 2014 Canadian Pensioner’s

Mortality Table3 (CMP2014) for a female pensioner, without mortality improvements. We

consider that T represents the time at which the plan member reaches 65. Therefore, for

T = 20, we use the probability that a woman aged 45 at t = 0 is still alive at 65 years old. In

that case, we have P (τ > 20) ≈ 0.9539. In the examples where T varies, we vary the age of the

plan member at t = 0 accordingly.

4.1 Initial cost of the hedging strategy

Figure 1a confirms the analysis of Section 3.1.1 concerning the cost of the hedging strategy

as T increases. The presence of equity in the market has a more significant impact when the

maturity of the payoff is longer. Figure 1b shows that, as expected, the cost of the hedging

strategy increases to E∗[ζ] as ε goes to 0, whether equity is present in the market or not. It

also is interesting to note that with the parameters we use, the initial cost of the strategy in the

mixed market remains significantly lower than its bond-only counterpart, even for small values

of ε.

Figure 1c indicates that the initial cost of the strategy is slightly more sensitive to the probability

of survival when a risky asset is available. Nonetheless, for P (τ > T ) ≥ 0.8, the probability of

survival does not have an important impact on the initial cost. Note that we only tested values

above 0.8, which corresponds to the probability of a 55 year old woman surviving 25 years,

according to the CPM2014.

Further numerical analysis shows that the initial cost of the quantile hedging strategy has a

similar behaviour for various values of g and n, including the zero-coupon bond case n = 0.

Therefore, we do not illustrate these results here.

4.1.1 Unhedged payoff

As explained in Section 3.1.2, the impact of equity on the size and the limiting behaviour of

the expected loss depends on the size of the market price of interest rate risk θr. Using the

parameters in Table 1, we have κ = 0.01269 < θr. Therefore, the presence of equity leads to an

increase in the expected loss. In both cases, the expected loss approaches 0 as ε→ 0.

Figure 2 presents the sensitivity of the expected loss to the market price of equity risk θ, as well

as its asymptotic behaviour when ε goes to 0. We illustrate the expected loss as a percentage

of the superhedging price of the claim, for comparison purposes. While a higher market price of

equity risk leads to a lower initial hedging cost, it is also associated with a higher expected loss,

if loss occurs. One can consider this higher expected loss as the price to pay for a less expensive

hedge. It also points out a flaw in the optimization criteria. In fact, by focusing only on the

3The 2014 Canadian Pensioner’s Mortality Table is available at http://www.cia-ica.ca/docs/

default-source/2014/214013e.pdf.

20

0 5 10 15 20 25 30

0.8

00.8

50.9

00.9

51.0

0

T

Initia

l cost of str

ate

gy (

as %

of superh

edgin

g p

rice)

Mixed market

Bond−only

(a) ε = 0.01

0.990 0.992 0.994 0.996 0.998 1.000

0.8

50.9

00.9

51.0

0

1 − ε

Initia

l cost of str

ate

gy (

as %

of superh

edgin

g p

rice)

Mixed market quantile hedging

Bond−only quantile hedging

Superhedge

(b) T = 20, P (τ > T ) = 0.9539

0.80 0.85 0.90 0.95 1.00

0.8

50.9

00.9

51.0

0

P(τ > T)

Initia

l cost of str

ate

gy (

as %

of superh

edgin

g p

rice)

Mixed market

Bond−only

(c) ε = 0.01, T = 20

Figure 1: Sensitivity of the initial cost of the strategy to (a) maturity T of the payoff, (b)

probability of loss ε and (c) survival probability P (τ > T ).

probability of success, the optimal hedging strategy ignores the size of the loss that occurs with

probability ε. Not controlling the size of this loss can represent an important risk for the hedger.

The behaviour of E[L|L > 0] → 0 as ε → 0 is illustrated, albeit not perfectly, in Figure 2b.

Although the limit of the expected loss is 0 in both markets, convergence is very slow and hard

to illustrate numerically. This especially true in the mixed market model, where the expected

loss stays above 80% of the superhedging price of the payoff for values of ε as small as 0.005.

To explore the behaviour of the expected loss when θr < κ, we also let θr = 0.01. This

has two effects: first, the expected loss is now lower in the mixed market than in its bond-only

counterpart, and second, the expected loss increases as the probability of success 1−ε approaches

21

0.0 0.1 0.2 0.3 0.4

0.9

20.9

40.9

60.9

8

θ

Expecte

d L

oss (

as %

of superh

edgin

g p

rice)

Mixed market

Bond−only

(a) ε = 0.01

0.95 0.96 0.97 0.98 0.99 1.00

0.8

50.9

00.9

5

1 − ε

Expecte

d L

oss (

as %

of superh

edgin

g p

rice)

Mixed market

Bond−only

(b) θ = 0.24

Figure 2: Sensitivity of E[L|L > 0] to θ (left) and to the probability of loss ε (right).

1, as predicted by Proposition 3.6. This is illustrated in Figure 3a.

Since κ is a function of T , among others, changes in the maturity of the payoff will affect the

behaviour of the expected loss. Figure 3b shows the conditional expected loss for T ∈ [1, 40]

when ε = 0.01. For lower values of T , the expected loss is lower in the bond-only market, but

the opposite becomes true for large enough T .

0.95 0.96 0.97 0.98 0.99 1.00

1.0

01.0

21.0

41.0

61.0

81.1

01.1

2

1 − ε

Expecte

d L

oss (

as %

of superh

edgin

g pri

ce)

Mixed market

Bond−only

(a) T = 20

0 10 20 30 40

1.0

01.0

51.1

01.1

51.2

01.2

51.3

0

T

Expecte

d L

oss (

as %

of superh

edgin

g p

rice)

Mixed market

Bond−only

(b) ε = 0.01

Figure 3: Sensitivity of E[L|L > 0] to the probability of loss ε (left) and to the maturity of the

payoff T (right) when θr = 0.01.

For the final part of this section, we return to the original parameter set presented in Table 1.

We consider different crediting rates by changing g and n. The conditional expected loss for

22

these two new payoffs is presented in Figure 4. On the left-hand side, we consider the expected

loss resulting from a quantile hedging strategy for a zero-coupon bond with 20 years to maturity

(that is, n = g = 0). On the right-hand side, the payoff considered is the accumulation of one

unit of currency at the constant rate g = 0.01 (keeping n = 0). In both cases, as in Figure

2b, the mixed market expected loss is higher than its bond market counterpart. However, as a

percentage of the superhedging price of the payoff, the expected loss is slightly higher when the

crediting rate is no longer linked to the term structure (n = 0.) This can be explained by the

lack of correlation between the payoff, which is now deterministic, and the bond market.

0.95 0.96 0.97 0.98 0.99 1.00

0.5

0.6

0.7

0.8

0.9

1.0

1 − ε

Expecte

d L

oss (

as %

of superh

edgin

g pri

ce)

Mixed market

Bond−only

(a) g = 0, n = 0

0.95 0.96 0.97 0.98 0.99 1.00

0.5

0.6

0.7

0.8

0.9

1.0

1 − ε

Expecte

d L

oss (

as %

of superh

edgin

g pri

ce)

Mixed market

Bond−only

(b) g = 0.01, n = 0

Figure 4: Sensitivity of E[L|L > 0] to the probability of loss ε, T=20.

5 Concluding remarks

In this paper, we applied the concept of quantile hedging introduced in (Follmer and Leukert,

1999) to a particular type of pension payoff, which is linked to interest rates, taking into account

survival of the plan member. Our goal was to assess the impact of the availability of equity on

the cost of the quantile hedging strategy, and therefore to highlight the incentive resulting from

a hedging criterion based solely on the probability of loss at maturity. The analysis of the

performance of the quantile hedge was then used to illustrate the risk linked with the strategies

yielding the lowest cost. Using simple models to represent bond-only and bond-and-equity

markets allowed us to obtain explicit expressions for the cost of the optimal strategy. Thus, we

identified a cost reduction when equity is available. We further gave expressions for the expected

value of the unhedged loss, which we showed can increase when equity is used in the partial

hedging strategy.

The criterion that we use to find optimal hedging strategies only relies on the probability of

loss at maturity. Our numerical results show that this loss can be considerable, especially

23

when equity can be added to the hedging portfolio, given that the interest rate risk premium is

sufficiently high. However, including equity in the portfolio also reduces the cost of the hedge,

which represents an important incentive to invest part of the hedging portfolio in risky assets.

This result can be relevant in light of current or planned regulation based on the VaR risk

measure, which also only takes into account the probability of loss. Further research should

explore different efficient hedging criteria (for example, those presented in Barski (2016) and

in Cong, Tan, and Weng (2014)) in the context of long-term pension claims to assess whether

those modified criteria still encourage investment in equity by reducing the cost of the hedge.

In particular, the size of the hedge should be taken into account.

The results obtained here depend on the market model we consider. Assuming specific dynamics

for the short-rate and the risky asset allowed us to obtain explicit expressions that could be

analysed. Completeness of the financial market is also a strong assumption that should be

relaxed in future research. Adding incompleteness to the market through jumps in the risky

asset price, for example, might increase the cost of the strategy in the mixed market.

Finally, our explicit results concern the initial cost of the hedging strategy. We do not present

explicit expressions for the investment strategy. Because of the form of the payoff and of the

completeness of the financial market, it is however possible to obtain expressions for the optimal

quantile hedging strategy in each market.

A Proof of Proposition 2.1

A version of this result for a slightly different payoff H appears in Nakano (2011). The proof

that we give here follows their proof closely.

Proof. First, set x = E∗[Y ] and let ξ be the perfect replicating strategy for Y (this strategy

exists since Y ∈ FT ). Then, since

x+

∫ T

0ξsdXs = Y,

and since Y ≥ H, we have that x = E∗[Y ] ≥ Π(H), by definition of Π(·).Now consider some admissible strategy x, ξ with

V x,ξT = x+

∫ T

0ξsdXs ≥ H.

Then, for any δ ∈ D, we have

E[ZTZδTH] ≤ E[ZTZ

δTV

x,ξT ] ≤ x, (A.1)

where the second inequality comes from the supermartingale property of ZtZδt Vx,ξt . By taking

the infimum over x and by expressing the left-hand side of (A.1) as E[ZTZδTY 1τ>T], we can

write

supδ∈D

E[ZTZδTY 1τ>T] ≤ Π(H).

24

But for any constant δ > −1, we have

E[ZTZδTY 1τ>T] = E[ZTY (1 + δ1τ≤T)e

−δ∫ τ∧T0 λsds1τ>T]

= E[ZTY e−δ∫ T0 λsds1τ>T]

= E[ZTY ΛT e−δ∫ τ∧T0 λsds]

= E[ZTY e−(δ+1)

∫ τ∧T0 λsds].

Then, E[ZTZδTY 1τ>T] → E[ZTY ] as δ −1. It follows that E[ZTY ] = E∗[Y ] ≤ Π(H). This

ends the proof. 2

B Proof of Proposition 2.3

To derive the distribution of ZT ζ, we first need to present the following lemmas. In Lemma B.1,

we recall a result from stochastic calculus.

Lemma B.1. Let W (t)0≤t≤T be a standard Brownian motion with W (0) = 0, and fix 0 ≤ t ≤T . Conditional on the value of W (T ), W (t) follows a Gaussian distribution with

E[W (t)|W (T ) ∈ dy] =t

Ty, (B.1)

and

Cov(W (t),W (s)|W (T ) ∈ dy) = min(t, s)− ts

T. (B.2)

The result of Lemma B.1 can be obtained by first showing that the Brownian motion W (t)0≤t≤T ,

conditioned on its value at T ,4 has the same distribution as a Brownian bridge. This can be

proven by deriving the joint density of W (t1),W (t2), . . . ,W (tn),W (T ), with 0 ≤ t1 ≤ t2 ≤ . . . ≤tn ≤ T and hence the density of W (t1),W (t2), . . . ,W (tn) conditioned on the value of W (T ),

which coincides with the density of a Brownian bridge reaching the same value at T . The inter-

ested reader is referred to Section 4.7.5 of Shreve (2004) for further details on the correspondence

between the conditioned Brownian motion and the Brownian bridge. Once this correspondence

is established, the result of Lemma B.1 follows from the distribution of the Brownian bridge,

and can be found for example in Section IV.4 of Borodin and Salminen Borodin and Salminen

(1996). It can also be obtained by integrating directly over the joint distribution of W (s) and

W (t) conditioned on the value of W (T ). Below, we present a proof of (B.1), based on Section

4.7.5 of Shreve (2004). We omit the proof of (B.2), as it is lengthy and is not the main focus of

the paper.

Proof. Let p(t, x) = P (W (t) ∈ dx) denote the density of the Brownian motion at time t ∈ [0, T ],

and recall that

p(t, x) =1√2πt

e−x2

2t .

4It is important to note that here, we only condition on the value of the Brownian motion at time T , and not

on its entire path from 0 to T , nor on the filtration induced by Wt0≤t≤T .

25

Since the increments of the Brownian motion are independent and stationary, the joint density

of W (t) and W (T ) can be written as

P (W (t) ∈ dx,W (T ) ∈ dy) = p(t, x)p(T − t, y − x)

=1√

2πt(T − t)e− 1

2

(x2

t+

(y−x)2T−t

).

Using the joint distribution of W (t) and W (T ), the distribution of W (t) conditional on the value

of W (T ) can be obtained as follows.

P (W (t) ∈ dx|W (T ) ∈ dy) =P (W (t) ∈ dx,W (T ) ∈ dy)

P (W (T ) ∈ dy)

=

√T√

2πt(T − t)exp

−1

2

(x2

t+

(y − x)2

T − t− y2

T

)=

1√2π(t/T )(T − t)

exp

− (x− y(t/T ))2

2(t/T )(T − t)

.

Therefore, conditional on W (T ) ∈ dy, W (t) has a normal distribution with mean y tT , which is

given in (B.1), and variance t(T−t)T . 2

Lemma B.2. Let W (t)0≤t≤T be a standard Brownian motion with W (0) = 0 and denote

I(T ) =:∫ T

0 eatW (t)dt. Conditional on the value of W (T ), I(T ) follows a Gaussian distribution

with

E [I(T )|W (T ) ∈ dy] =: yµI|W

= −y eaT

aT(D(T )− T ),

and

V ar (I(T )|W (T ) ∈ dy) =: s2I|W

=e2aT

a2

(D(T )−D2(0, T )

(a

2+

1

T

)).

Proof. Using Lemma 2, we have that conditional on the value of W (T ), eatW (t) is Gaussian for

any t ∈ [0, T ], and the integral∫ T

0 eatW (t)dt is thus also Gaussian. Next, we obtain its expected

value.

E[∫ T

0eatW (t) dt

∣∣∣W (T ) ∈ dy]

=

∫ T

0eatE[W (t)|W (T ) ∈ dy] dt

=

∫ T

0

t eat

Ty dt

=y

T

eaT (aT − 1) + 1

a2

= −eaT y

aT(D(T )− T ).

26

Note that from (B.1) and (B.2), we have

E[W (t)W (s)|W (T ) ∈ dy] = min(t, s) + st

(( yT

)2− 1

T

). (B.3)

To obtain the variance of∫ T

0 eatW (t)dt, it remains to obtain the expected value of its square.

E

[(∫ T

0eatW (t) dt

)2∣∣∣∣∣W (T ) ∈ dy

]

= E

[(∫ T

0eatW (t) dt

)(∫ T

0easW (s) ds

) ∣∣∣∣∣W (T ) ∈ dy

]

=

∫ T

0

∫ T

0ea(s+t)E[W (t)W (s)|W (T ) ∈ dy]ds dt

Using (B.3) in the integral above and performing the relatively simple (but tedious) integration

yields

E

[(∫ T

0eatW (t) dt

)2∣∣∣∣∣W (T ) ∈ dy

]

=e2aT (2aT − 3) + 4eaT − 1

2a3− (eaT (aT − 1) + 1)2

a4T+ E

[∫ T

0eaTW (t)dt

∣∣∣∣W (T ) ∈ dy]2

.

It follows that

V ar

(∫ T

0eatW (t) dt

∣∣∣∣W (T ) ∈ dy)

= E

[(∫ T

0eatW (t) dt

)2∣∣∣∣∣W (T ) ∈ dy

]− E

[∫ T

0eaTW (t)dt

∣∣∣∣W (T ) ∈ dy]2

=e2aT (2aT − 3) + 4eaT − 1

2a3− (eaT (aT − 1) + 1)2

a4T.

The desired results follows from re-arranging, and by using D(T ) = 1−eaTa , following the notation

introduced for the zero-coupon bond price in Section 2.1.1. 2

Lemma B.3. Let Y = λ0

∫ T0 eatW (t)dt+ λ1W (T ) + λ2W (T ), where λ0, λ1 and λ2 are strictly

positive constants, and W (t) and W (t) are Brownian motions with W (0) = W (0) = 0. Then Y

follows a Normal distribution with mean 0 and variance

λ20(s2

I|W + µ2I|WT ) + 2λ0µI|Wλ1T + (λ2

1 + λ22)T,

where s2I|W and µ2

I|W are as defined in Lemma B.2.

Proof. This result can be shown by calculating the characteristic function of Y . Conditioning

on the value of W (T ) and using independence of W (t) and W (t), as well as the result given in

Lemma B.2, will yield the desired result. 2

27

We will now prove Proposition 2.3 using Lemmas B.2 and B.3.

Recall that ζ denotes the discounted purely financial payoff at maturity T . From (2.10) and

(2.8), we have

ZT ζ = exp

θ′W (T )− T

2θ′θ +

∫ T

0(rc(t;n)− r(t)) dt

= exp

η(n, T )− T

2θ′θ + D(n)σre

−aT∫ T

0eatWr(t)dt+ θ′W (T )

, (B.4)

where

η(n, T ) =

gT −γ(n)Tn + D(n)

((r0 − b)1−e−aT

a + bT), n > 0,

gT −(

(r0 − b)1−e−aTa + bT

), n = 0,

(B.5)

and

D(n) =

D(n)−n)

n , n > 0

−1, n = 0.

To obtain (B.4), we solve (2.1) (see for example Section 3.2 of Brigo and Mercurio (2007)) and

get

r(t) = e−at(r0 + b(eat − 1)−

∫ t

0σre

audWr(u)

).

Integrating and using Ito’s lemma gives∫ T

0r(t)dt = (r0 − b)

1− e−aT

a+ bT − σre−aT

∫ T

0eatWr(t)dt,

which we use along with the definition of yn(t) to get (B.4) and (B.5).

The result follows from using Lemma B.3 with λ0 = D(n)σre−aT and (λ1, λ2)′ = θ.

C Proof of Proposition 3.4

In order to prove Proposition 3.4, we first need the two following lemmas.

Lemma C.1. Let W (t)t≥0 and W (t)t≥0 be independent Brownian motions, and let

I(T ) =

∫ T

0eatW (t)dt.

Then, (νI(T ), νI(T ) + θ1W (T ) + θ2W (T )) has a bivariate Normal distribution with mean (0, 0)

and covariance matrix[ν2s2

I ν2s2I + νµI|W θ1T

ν2s2I + νµI|W θ1T ν2s2

I + 2νµI|W θ1T + (θ21 + θ2

2)T

],

where µI|W and s2I|W are as defined in Lemma B.2, and where s2

I = (s2I|W + µ2

I|WT ).

28

Proof. The joint distribution of (νI(T ), νI(T ) + θ1W (T ) + θ2W (T )) is obtained by calculating

the moment generating function (MGF):

E[eκ1νI(T )+κ2(νI(T )+θ1W (T )+θ2W (T ))

]= E

[E[e(κ1+κ2)νI(T )+κ2(θ1W (T )+θ2W (T ))|W (T ), W (T )

]]= e

12

(κ1+κ2)2ν2s2I|WE

[e((κ1+κ2)νµI|W+κ2θ1)W (T )

]E[eκ2θ2W (T )

]= exp

1

2(κ1 + κ2)2ν2s2

I|W +1

2((κ1 + κ2)νµI|W + κ2θ1)2T +

1

2κ2

2θ22T

= exp

1

2

(κ2

1ν2s2I + 2κ1κ2(ν2s2

I + θ1νµI|WT ) + κ22(ν2s2

I + 2νµI|W θ1T + (θ21 + θ2

2)T ))

The desired result is obtained by comparing the expression obtained to the MGF of a bivariate

normal distribution. 2

Lemma C.2. Let (X,Y ) be bivariate Normal random variables with parameters µ = (0, 0) and

Σ =

[σ2X ρσXσY

ρσXσY σ2Y

].

Let c1, c2 ∈ R be real constants. Then we have,

E[ec1X1Y >c2

]= e

12c21σ

2XΦ

(c1ρσXσY − c2

σY

).

Proof. The density of (X,Y ) is given by

fX,Y (x, y) =1

2πσXσY√

1− ρ2exp

− 1

2(1− ρ2)

(x2

σ2X

+y2

σ2Y

− 2ρxy

σXσY

).

This result follows by calculating the following integral:

E[ec1X1Y >c2

]=

∫ ∞c2

∫ ∞−∞

ec1xfX,Y (x, y)dx dy.

2.

A more general version of Lemma C.2 is presented in Theorem 4.1 of Melnikov and Romanyuk

(2008); see also Lemma 4.1 of Melnikov (2011).

We can now prove Proposition 3.4. By Proposition 2.1, the superhedging portfolio for HT = 1A2(k∗)ζ(1−NT )

is the replicating portfolio for 1A2(k∗)ζ. Therefore, we have

L = HT − VT =

ζ(1− 1A2(k∗)), τ > T

−ζ, τ ≤ T.

Then, the set on which L is strictly positive is exactly (A2(k∗))c ∩ τ > T, and it follows that

E[L|L > 0] =E[ζ(1− 1A2(k∗))1τ>T]

P (Ac2(k∗) ∩ τ > T)(C.1)

29

By Assumption 1, the survival process is independent of the financial market, and we have

P ((A2(k∗))c ∩ τ > T) = (1− P (A2(k∗)))P (τ > T ) = ε.

The last equality results from the definition of k∗ and the distribution of ZT ζ given by (3.2) and

Proposition 2.3, respectively.

Therefore, we can re-write (C.1) as

ε−1E[ζ(1− 1A2(k∗))1τ>T] = ε−1E[ζ(1− 1A2(k∗))1τ>T|τ > T ]P (τ > T )

= ε−1ΛTE[ζ(1− 1A2(k∗))1τ>T]

To obtain the expression presented in Proposition 3.4, it suffices to write HT as a function of

Ir(T ) =∫ T

0 eatWr(t)dt, and dP ∗

dP HT in terms of νIr(T ) + θ1Wr(T ) + θ2W (T ). The result of the

proposition is obtained using the distribution presented in Lemma C.1, and the expectation

given in Lemma C.2.

D Other useful results

Lemma D.1. Let Φ(·) denote the distribution of a standard Normal random variable, and let

z1−ε denote the 1− ε quantile, so that Φ−1(1− ε) = z1−ε. Then,

limε→0

Φ(a− z1−ε)

ε=

∞ a > 0,

1 a = 0,

0 a < 0.

Proof. First note that as ε goes to 0, z1−ε goes to infinity, so that limε→0 Φ(a− z1−ε) = 0. Using

l’Hopital’s rule, we have

limε→0

Φ(a− z1−ε)

ε= lim

ε→0φ(a− Φ−1(1− ε))[Φ−1]′(1− ε),

where φ(·) = Φ′(·), and [Φ−1]′(x) = 1φ(Φ−1(x))

. Therefore,

[Φ−1]′(1− ε) =1

φ(z1−ε),

and it follows that

limε→0

Φ(a− z1−ε)

ε

= limε→0

φ(a− z1−ε)

φ(z1−ε)

= limε→0

e−12

((a−z1−ε)2−z21−ε)

= limε→0

e−12a2+az1−ε

= limx→∞

e−12a2+ax.

The result follows from considering the three cases a < 0, a = 0 and a > 0. 2

30

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